DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-03

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Invertible Euclidean linear maps

Definition

Let $n\ge1$ . A linear map $A:\mathbb R^n\to\mathbb R^n$ (A linear map $L:\mathbb{R}^m\to\mathbb{R}^n$ in Euclidean coordinates) is invertible when there is a linear map $B:\mathbb R^n\to\mathbb R^n$ such that

$B(Au)=u\quad\hbox{and}\quad A(Bu)=u\qquad(u\in\mathbb R^n).$

The map $B$ is unique: if $C$ has the same two properties, then $C=C\circ(A\circ B)=(C\circ A)\circ B=B$ . It is denoted $A^{-1}$ .

Depends on

A linear map $L:\mathbb{R}^m\to\mathbb{R}^n$ in Euclidean coordinates

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Basic Analysis II, §8.2 Matrices and linear mappings (standard reference, not scraped)